
TL;DR
This paper demonstrates that avoiding a fixed pattern in strong rectangulations significantly reduces their exponential growth rate, with the proportion of such pattern-avoiding rectangulations tending to zero as size increases.
Contribution
It establishes the first uniform exponential upper bound for pattern-avoiding rectangulations using a novel pattern-insertion scheme and generating function analysis.
Findings
The growth constant for pattern-avoiding rectangulations is strictly less than 27/2.
Pattern avoidance causes an exponential drop in the number of rectangulations.
The proportion of pattern-avoiding rectangulations tends to zero as size grows.
Abstract
Fix a strong rectangulation pattern of size . We show that the growth constant of the class of strong rectangulations avoiding is strictly smaller than , the growth constant for all strong rectangulations. More precisely, forbidding any such yields a pattern-uniform exponential drop of at least . Consequently, the proportion of -avoiding rectangulations among all rectangulations tends to zero as . This is the first result on the uniform drop of exponential growth for pattern-avoiding rectangulations. The proof utilizes the standard correspondence with leftmost history quadrant walks, along with a pattern-insertion scheme that controls the radius of convergence of the associated generating functions, thereby establishing the first uniform exponential upper bound for rectangulation classes defined by geometric…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Computational Geometry and Mesh Generation · semigroups and automata theory
